Examples

Examples used in the Dictionary of Fine Distinctions. Each example is marked with the lowest level at which it is appropriate. Choosing a level shows the examples up to that level.

Elementary

• $1$ and $\setdef{1, 2}$
  Terms: subset; proper subset
• $3 - 7 = -4$
  Terms: difference; distance
  Distinctions: Types of difference
• $9.8 - 9.81 = -0.01$, for a measured value $9.8$ and true value $9.81$
  Terms: error; difference
  Distinctions: Types of difference
• A bar graph of monthly rainfall
  Terms: chart
• A clock that chimes every hour
  Terms: periodic; aperiodic
• A drawing of $y = x^2$ on coordinate axes
• A flowchart
• A line
  Terms: dimension (of a vector space)
• A pie chart
  Terms: chart
• A flat surface (a plane)
  Terms: dimension (of a vector space)
• A robot arm following a moving conveyor belt
  Terms: tracking
  Distinctions: Types of control objectives
• A scatter plot of height vs. weight
  Terms: chart
• A spacecraft following a planned trajectory
  Terms: tracking
  Distinctions: Types of control objectives
• A square
• A temperature that goes from $20$ to $15$ degrees
  Terms: change
  Distinctions: Types of difference
• A thermostat
  Terms: closed loop; feedback; open loop; feedforward
  Distinctions: Types of control
• A toaster that runs for a fixed time
  Terms: open loop; closed loop; feedback
  Distinctions: Types of control
• $\abs{3 - 7} = 4$
  Terms: distance
  Distinctions: Types of difference
• Addition $+$ on $\reals$
  Terms: binary operation; associative; commutative; relation
  Distinctions: Associative vs. commutative; Operation vs. relation
• Appending $4$ to $(1, 2, 3)$ produces $(1, 2, 3, 4)$
  Terms: append; prepend
• Two adjacent sides of a rectangle, as segments meeting at a corner
  Terms: right angle; perpendicular
  Distinctions: Right angle vs. perpendicular vs. orthogonal vs. normal
• Cruise control holding 100 km/h on hilly roads
  Terms: tracking
  Distinctions: Types of control objectives
• The data $1, 1, 2, 3, 100$
  Distinctions: Measures of center
• $\emptyset$ and $\setdef{1, 2}$
  Terms: subset; proper subset
• “I have a theory about who ate the cake”
  Terms: theory (in science)
  Distinctions: Conjecture vs. hypothesis vs. law vs. theory
• $-5$, in $-5x$
• Multiplication $\times$ on $\reals$
  Terms: binary operation; associative; commutative; relation
  Distinctions: Associative vs. commutative; Operation vs. relation
• The number $5$ under multiplication of real numbers
  Terms: inverse element
• $1$
  Terms: natural numbers (from 1); natural numbers (from 0); whole numbers
  Distinctions: Natural numbers: starting at 0 or at 1
• $\frac{1}{2}$ and $\frac{2}{4}$
  Terms: rational number; equal; equivalent
  Distinctions: Types of sameness
• A phone number
  Terms: cardinal number; ordinal number
• $\pi$
  Terms: irrational number; transcendental number; real number; rational number; integer; algebraic number
• Before an experiment: “Bean plants grow taller with more hours of light each day”
  Terms: hypothesis (in science); theory (in science); law (in science)
  Distinctions: Conjecture vs. hypothesis vs. law vs. theory
• $\setdef{1, 2, 2}$
  Terms: set; collection; multiset
  Distinctions: Types of collections
• $\setdef{1, 2, 3} \setminus \setdef{2} = \setdef{1, 3}$
  Terms: set difference; difference
  Distinctions: Types of difference
• $\setdef{1, 2}$ and $\setdef{1, 2}$
  Terms: subset; proper subset
• $\setdef{1}$ and $\setdef{\setdef{1}, 2}$
  Terms: subset; proper subset
• The set $\setdef{3, 5, 8}$
• $A = \setdef{1, 2}$ and $B = \setdef{3, 4}$
  Terms: union; intersection; set difference; disjoint union (of disjoint sets)
• Steering a bike to keep from tipping over
  Terms: feedback; closed loop
• Subtraction $a - b$ on $\reals$
  Terms: associative; commutative
  Distinctions: Associative vs. commutative
• The “23” on a jersey
  Terms: cardinal number; ordinal number
• The “3” in “3 apples”
  Terms: cardinal number; ordinal number
• The “3rd” in “finished 3rd”
  Terms: ordinal number; cardinal number
• The location of Paris
  Terms: point; vector
  Distinctions: Vector vs. point
• The sequence $1, 2, 3, \dots$
  Terms: sequence; bounded
  Distinctions: Types of infinity
• $2.5$
  Terms: natural numbers (from 0); whole numbers
  Distinctions: Whole numbers vs. natural numbers
• Walking $3$ m east and then $3$ m west
  Terms: displacement; distance
  Distinctions: Types of difference
• $0$
  Terms: natural numbers (from 0); whole numbers; zero; natural numbers (from 1); zero vector
  Distinctions: Natural numbers: starting at 0 or at 1; Whole numbers vs. natural numbers

High school

• $(0, 0) \in \reals^2$, for vector addition
  Terms: zero vector; identity element; zero
• In floating point arithmetic, $0.1 + 0.2$ evaluates to $0.30000000000000004$ due to rounding errors.
  Terms: error; bug
• $0$, for the set $S = (0, 1]$
• $0$, for $x \mapsto x^2$
  Terms: fixed point
• $0 < 0.5 < 1$, for the interval $[0, 1]$
• The list of numbers $(1, 3, -2.4)$, with no other information
• $1$, for $S = [1, 2]$
• $1$, for $\dot{x} = f(x) = 1 - x$
  Terms: steady state; fixed point
• $(2, 3) \in \reals^2$
  Terms: vector; point
  Distinctions: Vector vs. point
• $2$, for $f(x) = x^2 - 4$
  Terms: zero
• $(3, 2)$ is the coordinate vector of $(3, 2) \in \reals^2$, relative to the standard basis
  Terms: vector; coordinate vector
  Distinctions: Vector vs. coordinate vector
• $3$ and $10$, as integers modulo $7$
  Terms: equivalent; congruent; equal
  Distinctions: Types of sameness
• $3$, for $f(x) = x^2 + 3$
• $3x + 2$
• “$7$ is prime”
  Terms: proposition; propositional formula; relation; sentence
  Distinctions: Proposition vs. predicate; Proposition vs. propositional formula; Sentence vs. proposition
• $\reals^3$
  Terms: dimension (of a vector space)
• A continuous map from a sphere to the plane
  Distinctions: Function vs. map
• $A = \pi r^2$
• A gust of wind hitting a quadcopter drone
  Terms: exogenous input; disturbance; control input
  Distinctions: Types of exogenous inputs
• A heated rod whose ends lose heat to the air (Newton’s law of cooling)
• A heated rod with insulated ends
• A heated rod with its ends held in ice water
• A heated rod with one end in ice water and the other insulated
• A least-squares estimate of a constant parameter from data
  Terms: estimator; observer; filter (estimation); predictor
  Distinctions: Types of estimators
• A low-pass filter on a sensor signal
  Terms: observer; filter (estimation); predictor
  Distinctions: Types of estimators
• A payload that makes a quadcopter drone 10% heavier than modeled
  Terms: exogenous input
  Distinctions: Types of exogenous inputs
• A precomputed optimal input $u(t)$, applied without measurements
  Terms: open loop; feedback
  Distinctions: Types of control
• A solid bowling ball
  Terms: ball; sphere
• A thermostat setting of 20 degrees
• A thermostat that switches a heater on and off
  Terms: switched system
• The probability that a uniform random number on $[0, 1]$ is rational
  Terms: probability zero
  Distinctions: Types of set sparsity
• A velocity, in meters per second
  Terms: vector; covector
  Distinctions: Vector vs. covector
• A weather forecast for tomorrow
  Terms: predictor
  Distinctions: Types of estimators
• $f(x) = \abs{x}$
  Terms: differentiable; continuously differentiable; smooth
  Distinctions: Convex set vs. convex function; Differentiable vs. continuously differentiable
• The absolute value of a number
• An aircraft approaching a tanker aircraft and then holding position behind the tanker to refuel in flight
  Terms: tracking
• The ring $\setdef{(x, y) \suchthat 1 \leq x^2 + y^2 \leq 4}$ (an annulus)
  Distinctions: Convex set vs. convex function
• The average $a \ast b = \frac{a + b}{2}$ of two real numbers
  Terms: commutative; associative
  Distinctions: Associative vs. commutative
• Balancing an inverted pendulum upright
  Terms: stabilization; tracking
  Distinctions: Types of control objectives
• The matrix \(\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}\)
  Terms: eigenvalue; singular value; array; matrix; linear map; symmetric matrix
  Distinctions: Array vs. matrix vs. linear map; Eigenvalue vs. singular value
• Choosing a committee of $2$ members from a club of $10$ members
• The closed unit disk $D = \setdef{(x, y) \suchthat x^2 + y^2 \leq 1}$
  Terms: interior; boundary; exterior; area; diameter; cardinality; measure; Hausdorff dimension
  Distinctions: Convex set vs. convex function
• $[0, 1]$
  Terms: arc length; diameter; cardinality; measure; Hausdorff dimension; interior; boundary; exterior; closed set; bounded; infinite; uncountable; open set; finite
  Distinctions: Types of infinity
• $x \mapsto 5$
• Cruise control that uses both the speed error and a map of upcoming hills
  Terms: closed loop; feedforward; feedback
  Distinctions: Types of control
• The empty set $\emptyset$
  Terms: open set; closed set; finite; uncountable
• $=$, $<$, and $\subseteq$
  Terms: relation; binary operation
  Distinctions: Operation vs. relation
• $d(x, y) = \sqrt{(x_1 - y_1)^2 + (x_2 - y_2)^2}$ on $\reals^2$

  The Euclidean distance between points in the plane.

  Terms: metric
  Distinctions: Norm vs. metric vs. inner product
• The Euclidean norm $x \mapsto \sqrt{x_1^2 + x_2^2}$ on $\reals^2$

  The Euclidean norm, which gives the length of a vector in the plane.

  Terms: norm; covector
  Distinctions: Norm vs. metric vs. inner product; Vector vs. covector
• $e$
  Terms: transcendental number; irrational number; real number
• $f \mapsto f^{\prime}$, on differentiable functions
  Terms: operator; functional
• The roll of a fair die
  Distinctions: Measures of center
• Goldbach’s conjecture
  Terms: hypothesis (in science)
  Distinctions: Conjecture vs. hypothesis vs. law vs. theory
• Great-circle distance on the surface of the Earth
  Terms: metric; norm
  Distinctions: Norm vs. metric vs. inner product
• $[0, 1)$, as a subset of $\reals$
  Terms: interior; boundary; exterior; open set; closed set
• The $3 \times 3$ identity matrix
  Terms: shape; positive definite matrix; positive semidefinite
  Distinctions: Positive definite vs. positive semidefinite
• $x \mapsto x$, for composition of functions
  Terms: identity element
• The integers $\mathbb{Z}$
  Terms: infinite; finite; uncountable
• $\mathbb{Z}$ with addition
  Terms: group
  Distinctions: Types of collections
• The interval $(0, 1]$
• The function $f(x) = 2x + 1$ on $\reals$ under composition of functions
  Terms: inverse element
• Jitter in a quadcopter drone’s GPS position readings
  Terms: exogenous input; disturbance; control input
  Distinctions: Types of exogenous inputs
• The lines $y = x$ and $y = -x$ in the plane, with direction vectors $(1, 1)$ and $(1, -1)$
  Terms: right angle; perpendicular; orthogonal
  Distinctions: Right angle vs. perpendicular vs. orthogonal vs. normal
• $\nats$
  Terms: infinite; finite; bounded; uncountable
  Distinctions: Types of infinity
• $\nats$ with addition
  Terms: group
  Distinctions: Types of collections
• $f(x) = -\abs{x}$
• Newton’s law of universal gravitation, $F = G m_1 m_2 / r^2$
  Terms: law (in science); theory (in science); hypothesis (in science)
  Distinctions: Conjecture vs. hypothesis vs. law vs. theory
• The north pole $p = (0, 0, 1)$ and the point $q = (1, 0, 0)$ on the equator of the unit sphere in $\reals^3$
  Terms: Euclidean distance; Taxicab distance; Chebyshev distance; discrete metric; geodesic distance
  Distinctions: Types of distances
• The $1000 \times 1000$ matrix of ones
• The $3 \times 4$ matrix of ones
  Terms: shape
• $(0, 1)$
  Terms: interior; boundary; exterior; open set; bounded; closed set; finite
• In Python, opening a file that does not exist raises FileNotFoundError
  Terms: bug
• $P \implies Q$ and $\neg Q \implies \neg P$
  Terms: equivalent; equal
  Distinctions: Types of sameness
• The parabola $y = x^2$
• Euclid’s parallel postulate
  Terms: postulate; axiom
• The vectors $(1, 0)$ and $(2, 0)$ in $\reals^2$
  Terms: basis; orthogonal; pairwise linearly independent
  Distinctions: Linearly independent vs. orthogonal; Linearly independent vs. pairwise non-parallel
• The swinging pendulum of a grandfather clock
  Terms: periodic
• A pendulum swinging on its own
• The points $p = (0, 0)$ and $q = (100, 100)$ in $\reals^2$
  Terms: Euclidean distance; Taxicab distance; Chebyshev distance; discrete metric; geodesic distance
  Distinctions: Types of distances
• The points $p = (0, 0)$ and $q = (2, 0)$ in $\reals^2$
  Terms: Euclidean distance; Taxicab distance; Chebyshev distance; discrete metric; geodesic distance
  Distinctions: Types of distances
• The points $p = (1, 1)$ and $q = (4, 5)$ in $\reals^2$
  Terms: Euclidean distance; Taxicab distance; Chebyshev distance; geodesic distance; discrete metric
  Distinctions: Types of distances
• $x^2 - 1 = 0$
• $f(x) = x^2 - 1$, for $x \in \reals$
  Terms: polynomial function
• $x^2 - 1$
  Terms: polynomial function
• Raising heater power when an outdoor thermometer reports a cold snap
  Terms: feedforward; feedback
  Distinctions: Types of control
• The set of rational numbers, $\mathbb{Q}$
  Terms: measure zero; nowhere dense; uncountable
  Distinctions: Types of set sparsity
• $\mathbb{Q} \cap [0, 1]$
  Terms: diameter; cardinality; measure; Hausdorff dimension; infinite; bounded; uncountable; finite
  Distinctions: Types of infinity
• The set of real numbers, $\reals$
  Terms: diameter; cardinality; measure; Hausdorff dimension; open set; closed set; uncountable; finite
• Rotation of the plane by $90°$
  Terms: linear map; matrix; array
  Distinctions: Array vs. matrix vs. linear map
• The set $\setdef{0, 1}$, as a subset of $\reals$
  Terms: diameter; cardinality; measure; Hausdorff dimension
• $\setdef{1, 2, 3}$
  Terms: cardinality; finite; bounded; sequence; uncountable
• $\setdef{(x, x^2) \suchthat x \in \reals}$
• $A = \setdef{\text{apple}}$ and $B = \setdef{\text{banana}}$
  Terms: union; disjoint union (of disjoint sets); disjoint union (tagged)
  Distinctions: Disjoint union: union of disjoint sets vs. tagged union
• $A = \setdef{\text{apple}, \text{orange}}$ and $B = \setdef{\text{orange}, \text{banana}}$
  Terms: union; intersection; disjoint union (of disjoint sets); disjoint union (tagged)
  Distinctions: Disjoint union: union of disjoint sets vs. tagged union
• $\sin^2 x + \cos^2 x = 1$
• $f(x) = \frac{\sin x}{x}$, at $x = 0$
  Terms: limit (of a function); left limit; right limit; continuous
• The vectors $(1, 0)$ and $(1, 1)$ in $\reals^2$
  Terms: basis; pairwise linearly independent; orthogonal
  Distinctions: Linearly independent vs. orthogonal
• The $x$-axis and the line through $(0, 0, 1)$ parallel to the $y$-axis in $\reals^3$
  Terms: orthogonal; perpendicular; right angle
  Distinctions: Right angle vs. perpendicular vs. orthogonal vs. normal
• $\sqrt{2}$
  Terms: irrational number; algebraic number; real number; integer; rational number; transcendental number
• The vectors $(1, 0)$ and $(0, 1)$ in $\reals^2$
  Terms: basis; orthogonal; pairwise linearly independent
  Distinctions: Linearly independent vs. orthogonal; Linearly independent vs. pairwise non-parallel
• Subtraction on $\nats$
  Terms: binary operation; relation
  Distinctions: Operation vs. relation
• The boundary of a square
  Terms: circle
• The data $1, 2, 3, 4$
  Distinctions: Variance vs. sample variance
• The motor commands of a quadcopter drone
  Terms: exogenous input; control input; disturbance
  Distinctions: Types of exogenous inputs
• The polynomial $p(x) = 3 + 2x$
  Terms: vector; coordinate vector
  Distinctions: Vector vs. coordinate vector
• The roots of $(x - 1)^2$ counted with multiplicity
  Terms: multiset; set
  Distinctions: Types of collections
• The row vector $\begin{bmatrix} 1 & 2 & 3 \end{bmatrix}$
  Terms: covector
  Distinctions: Vector vs. covector
• The surface of a basketball
  Terms: sphere; ball; open ball
• The surface of a cube
  Terms: sphere
• The temperature gradient, in degrees per meter
  Terms: covector
  Distinctions: Vector vs. covector
• The unit circle, for the closed unit disk in $\reals^2$
  Terms: boundary; edge (of a polygon or polyhedron); edge (of a graph)
• The theory of evolution by natural selection
  Terms: theory (in science); hypothesis (in science); law (in science)
  Distinctions: Conjecture vs. hypothesis vs. law vs. theory
• The vectors $(1, 0)$, $(0, 1)$, and $(1, 1)$ in $\reals^2$
  Terms: pairwise linearly independent; basis; orthogonal
  Distinctions: Linearly independent vs. pairwise non-parallel
• $G(s) = \frac{1}{s + 2}$
  Terms: pole; zero (of a transfer function)
• $G(s) = \frac{s - 1}{(s + 2)(s + 3)}$
  Terms: pole; zero (of a transfer function)
• $G(s) = \frac{s + 1}{(s + 1)(s + 2)}$
  Terms: pole; zero (of a transfer function)
• $(1, 2, 2)$
  Terms: sequence; indexed family; set; multiset
  Distinctions: Types of collections
• Two $3$-$4$-$5$ triangles in different places
  Terms: congruent; equivalent; equal
  Distinctions: Types of sameness
• $[0, 1] \cup [2, 3]$
• $u = -K x$, with the full state measured
  Terms: state feedback; output feedback
  Distinctions: State feedback vs. output feedback
• $u = -K y$, when $y = x$
  Terms: state feedback; output feedback
  Distinctions: State feedback vs. output feedback
• $\cup$ on the subsets of a set $X$
  Terms: binary operation; relation
  Distinctions: Operation vs. relation
• The vector $x = (1, 1) \in \reals^2$
  Terms: Euclidean norm; 1-norm; infinity norm
  Distinctions: Types of norms
• The vector $(1, 1)$ and the line $x + y = 1$ in the plane, whose direction vector is $(1, -1)$
  Terms: normal vector; orthogonal; perpendicular
  Distinctions: Right angle vs. perpendicular vs. orthogonal vs. normal
• The vector $x = (3, -4) \in \reals^2$
  Terms: Euclidean norm; 1-norm; infinity norm
  Distinctions: Types of norms
• The vector $x = (5, 0) \in \reals^2$
  Terms: Euclidean norm; 1-norm; infinity norm
  Distinctions: Types of norms
• The vectors $(2, 0)$ and $(0, 2)$ in $\reals^2$
  Terms: pairwise linearly independent; orthogonal
• $x = 0$, for $f(x) = x^2 + 3$
• “$x > 3$”
  Terms: predicate; proposition; relation; sentence; propositional formula; term
  Distinctions: Formula vs. sentence; Proposition vs. predicate; Sentence vs. proposition; Sentence vs. propositional formula; Term vs. formula
• $f(x) = x^2$ on $\reals$

  The squaring function, as a function $f : \reals \to \reals$.

  Terms: preimage; continuously differentiable; continuous; positive definite function; locally Lipschitz continuous; Lipschitz continuous; uniformly continuous; class-$\mathcal{K}$; operator; functional; transform
  Distinctions: Lipschitz continuous vs. continuously differentiable; Convex set vs. convex function; Function vs. map; Types of continuity
• $x^2 + 1$
  Terms: term; proposition; predicate; sentence
  Distinctions: Formula vs. sentence; Term vs. formula
• $x \mapsto 2x$, as a linear map $\reals \to \reals$
  Terms: injective; surjective
• $x \mapsto \sin x$
  Terms: differentiable; continuously differentiable; smooth; polynomial function
  Distinctions: Convex set vs. convex function; Differentiable vs. continuously differentiable
• $x \mapsto x \abs{x}$
  Terms: differentiable; continuously differentiable; smooth
  Distinctions: Differentiable vs. continuously differentiable
• $x \mapsto \abs{x}$ on $[-1, 1]$
  Terms: locally Lipschitz continuous; Lipschitz continuous; uniformly continuous; absolutely continuous; continuously differentiable
  Distinctions: Lipschitz continuous vs. continuously differentiable
• $x \mapsto x^2$ on $[0, 1]$
  Terms: locally Lipschitz continuous; Lipschitz continuous; continuously differentiable; absolutely continuous; continuous; uniformly continuous
  Distinctions: Lipschitz continuous vs. continuously differentiable; Types of continuity
• $\dot{x} = -x^3$
  Terms: asymptotically stable
• $\dot{x} = -x + \sin t$
• $\dot{x} = -x + u$
• $\dot{x} = x + 0 \cdot u$
  Terms: reachable

Undergraduate

• $(3, 2)$ is the coordinate vector of $3 + 2x$, relative to the basis $(1, x)$
  Terms: coordinate vector
  Distinctions: Vector vs. coordinate vector
• $(3, -1)$ is the coordinate vector of $3 + 2x$, relative to the basis $(1 + x, x)$
  Terms: coordinate vector
  Distinctions: Vector vs. coordinate vector
• “$7$ is prime and $4$ is not prime”
  Terms: proposition; propositional formula
  Distinctions: Proposition vs. propositional formula
• $\reals^2$ and $\mathbb{C}$, as real vector spaces
  Terms: isomorphic; identified; equivalent; equal
  Distinctions: Types of sameness
• A bouncing ball
  Terms: switched system
• A Kalman filter
  Terms: observer; estimator; filter (estimation); predictor
  Distinctions: Types of estimators
• A linear map $T : \reals^2 \to \reals^2$
  Terms: linear map
  Distinctions: Function vs. map
• A Luenberger observer
  Terms: observer; estimator; predictor
  Distinctions: Types of estimators
• A mass $m$ on a frictionless spring with stiffness $k$, at position $q$
  Terms: conservative system
• A particle filter
  Terms: estimator; filter (estimation); predictor
  Distinctions: Types of estimators
• A sampled-data control system
  Terms: switched system
• An extended Kalman filter
  Terms: estimator; filter (estimation); observer; predictor
  Distinctions: Types of estimators
• The matrix \(\begin{bmatrix} 1 & 0 & 2 \\ 0 & 1 & 3 \end{bmatrix}\)
  Terms: eigenvalue; singular value
  Distinctions: Eigenvalue vs. singular value
• The statement “$f$ is continuous on $[a, b]$,” in the intermediate value theorem
  Terms: axiom; postulate
• Customers arriving at a shop independently of each other at an average rate of $3$ per hour
  Terms: uniform distribution; Bernoulli distribution; Poisson distribution; exponential distribution; normal distribution
  Distinctions: Poisson vs. exponential distribution
• $d(x, y) = 1$ for $x \neq y$, and $d(x, x) = 0$
  Terms: metric; discrete metric; norm
  Distinctions: Norm vs. metric vs. inner product
• The matrix \(\begin{bmatrix} 2 & 0 \\ 0 & -3 \end{bmatrix}\)
  Terms: eigenvalue; singular value
  Distinctions: Eigenvalue vs. singular value
• Differentiation, on polynomials of degree at most $2$
  Terms: linear map; matrix
  Distinctions: Array vs. matrix vs. linear map
• $\dot x_1 = x_2$, $\dot x_2 = u$, with position output $y = x_1$
  Terms: observable; reconstructible; detectable
  Distinctions: Observable vs. reconstructible vs. detectable
• $\dot x_1 = x_2$, $\dot x_2 = u$, with velocity output $y = x_2$
  Terms: observable; reconstructible; detectable
  Distinctions: Observable vs. reconstructible vs. detectable
• An eigendecomposition $A = P D P^{-1}$, with $D$ diagonal
  Terms: singular value decomposition
  Distinctions: Types of matrix decompositions
• Emails arriving independently of each other at an average rate of $6$ per hour, and the event that no email arrives in the next $10$ minutes
  Terms: Poisson distribution; exponential distribution
  Distinctions: Poisson vs. exponential distribution
• Energy $H = \frac{1}{2}(p^2 + q^2)$, for the oscillator $\dot{q} = p$, $\dot{p} = -q$
  Terms: invariant
• Estimating the probability $p$ of heads for a coin from $7$ heads in $10$ independent flips
• $\exists x \, (x^2 = 2)$
  Terms: proposition; sentence
  Distinctions: Sentence vs. proposition
• $\exists y \, (y > x)$
  Terms: sentence
  Distinctions: Formula vs. sentence
• $f \mapsto f(0)$
  Terms: functional; operator
• $f(x) = x^2 \sin(1/x)$ for $x \neq 0$, $f(0) = 0$
  Terms: differentiable; continuously differentiable; smooth
  Distinctions: Differentiable vs. continuously differentiable
• $f(x) = x^2 \sin(1/x)$ for $x \neq 0$, $f(0) = 0$, on $[-1, 1]$
  Terms: continuous; Lipschitz continuous; uniformly continuous; absolutely continuous; continuously differentiable
  Distinctions: Lipschitz continuous vs. continuously differentiable; Types of continuity
• $f(x) = x^2$, $g(x) = x$

  Comparing $f$ with $g$ as $x \to \infty$.

  Terms: asymptotically equivalent
• Finding the shortest curve $y(x)$ in the plane from $(0, 0)$ to $(1, 1)$
  Terms: calculus of variations; optimal control
  Distinctions: Calculus of variations vs. optimal control
• $\forall x \, (x^2 \geq 0)$, with $x$ ranging over $\reals$
  Terms: sentence; proposition; predicate; propositional formula; term
  Distinctions: Formula vs. sentence; Proposition vs. predicate; Sentence vs. proposition; Sentence vs. propositional formula; Term vs. formula
• An invertible linear map $V \to V$
  Terms: linear map
• The relation $<$ on $\reals$
  Terms: relation; binary operation
• A linear map $\reals^2 \to \reals^3$
  Terms: linear map
• The matrix \(\begin{bmatrix} 2 & 1 \\ 4 & 3 \end{bmatrix}\)
  Terms: LU decomposition; Jordan–Chevalley decomposition; PLU decomposition; singular value decomposition; Cholesky decomposition; eigendecomposition; Schur decomposition; Jordan normal form; Toeplitz decomposition; symmetric matrix
  Distinctions: Types of matrix decompositions
• The matrix \(\begin{bmatrix} 3 & 1 \\ -1 & 1 \end{bmatrix}\)
  Terms: Jordan normal form; eigenvalue
• The matrix \(\begin{bmatrix} 4 & 2 \\ 2 & 3 \end{bmatrix}\)
  Terms: LU decomposition; PLU decomposition; Cholesky decomposition; eigendecomposition; singular value decomposition; Schur decomposition; Jordan normal form; Jordan–Chevalley decomposition; Toeplitz decomposition; symmetric matrix; positive definite matrix; positive semidefinite
  Distinctions: Positive definite vs. positive semidefinite; Types of matrix decompositions
• The matrix \(\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}\)
  Terms: eigenvalue; singular value; LU decomposition; QR decomposition; Jordan–Chevalley decomposition; PLU decomposition; singular value decomposition; Cholesky decomposition; eigendecomposition; Schur decomposition; Jordan normal form; Toeplitz decomposition
  Distinctions: Eigenvalue vs. singular value; Types of matrix decompositions
• Multiplication of $2 \times 2$ real matrices
  Terms: associative; commutative
  Distinctions: Associative vs. commutative
• The matrix \(A = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\)
  Terms: LU decomposition; QR decomposition; singular value decomposition; Jordan–Chevalley decomposition; PLU decomposition; Cholesky decomposition; eigendecomposition; Schur decomposition; Jordan normal form; Toeplitz decomposition
  Distinctions: Types of matrix decompositions
• Moving a cart from rest at position $0$ to rest at position $1$ in the shortest time, with an acceleration $u(t)$ between $-1$ and $1$
  Terms: optimal control; calculus of variations
  Distinctions: Calculus of variations vs. optimal control
• The open sets of $\reals$ (the unions of open intervals)
  Terms: topology; σ-algebra
  Distinctions: Topology vs. σ-algebra
• $p \land \neg q$
  Terms: propositional formula; proposition; predicate; sentence
  Distinctions: Proposition vs. propositional formula; Sentence vs. propositional formula
• The tautology $p \lor \neg p$
  Terms: propositional formula; proposition
  Distinctions: Proposition vs. propositional formula
• $f(x, y, z) = x^2 + y^2$
  Terms: gradient; Jacobian
  Distinctions: Types of derivatives in vector calculus
• $\setdef{\emptyset, \setdef{1}, \setdef{2, 3}, \setdef{1, 2, 3}}$, on $X = \setdef{1, 2, 3}$
  Terms: topology; σ-algebra
  Distinctions: Topology vs. σ-algebra
• The set of polynomials of degree at most $2$
  Terms: dimension (of a vector space)
• All subsets of a set $X$ (its power set)
  Terms: topology; σ-algebra
  Distinctions: Topology vs. σ-algebra
• The set of all subsets of $\nats$
  Terms: infinite; uncountable; finite
• $\mathbb{Q}$, as a subset of $\reals$
  Terms: interior; boundary; exterior; open set; closed set
• $\reals^n$ with the Euclidean norm
  Terms: metric space; Banach space
• $\reals$ with the standard topology
  Terms: T₁
• $(x, y) \mapsto x y + 1$ on $\reals^2$
  Terms: bilinear form; inner product
• $\setdef{\emptyset, \setdef{1}, \setdef{1, 2}}$, on $X = \setdef{1, 2}$
  Terms: topology; σ-algebra
  Distinctions: Topology vs. σ-algebra
• The functions $\sin$ and $\cos$ on $[-\pi, \pi]$, with the $L^2$ inner product $\ip{f}{g} = \int_{-\pi}^{\pi} f(x) g(x) \, dx$
  Terms: orthogonal; perpendicular
  Distinctions: Right angle vs. perpendicular vs. orthogonal vs. normal
• $f(x) = \frac{\sin x}{x}$ for $x \neq 0$, $f(0) = 0$, at $x = 0$
• The argument “All humans are mortal; Socrates is human; therefore, Socrates is mortal”
• The step function: $0$ for $x < 0$ and $1$ for $x \geq 0$
  Terms: limit (of a function); left limit; right limit; continuous
• $f(t, x) = t x^2$
  Terms: partial derivative; differential; total derivative (linear map); ordinary derivative
• The Cantor function on $[0, 1]$
  Terms: uniformly continuous; continuous; Lipschitz continuous; absolutely continuous; continuously differentiable
  Distinctions: Types of continuity
• The Cantor set
  Terms: diameter; cardinality; measure; Hausdorff dimension; closed set; nowhere dense; measure zero; uncountable; open set; finite
  Distinctions: Types of set sparsity
• The plane without the origin, for the Van der Pol oscillator
• The unit circle, for $\dot{x} = y$, $\dot{y} = -x$ (the harmonic oscillator)
  Distinctions: Periodic orbit vs. limit cycle
• Tricomi equation $y u_{xx} + u_{yy} = 0$
• $u = -K \hat{x}$, where an observer computes $\hat{x}$ from $y$
  Terms: output feedback; state feedback
  Distinctions: State feedback vs. output feedback
• $u = -k y$, for $\dot x_1 = x_2$, $\dot x_2 = u$ with $y = x_1$
  Terms: output feedback; state feedback
  Distinctions: State feedback vs. output feedback
• The vectors $(1, 0)$ and $(0, 0)$ in $\reals^2$
  Terms: orthogonal; perpendicular; pairwise linearly independent
  Distinctions: Linearly independent vs. orthogonal; Linearly independent vs. pairwise non-parallel; Right angle vs. perpendicular vs. orthogonal vs. normal
• Wave equation $u_{tt} = u_{xx}$
• $x_{k+1} = 0 \cdot x_k + 0 \cdot u_k$
  Terms: reachable
• $x_{k+1} = 0 \cdot x_k$, $y_k = 0$
  Terms: reconstructible; detectable; observable
  Distinctions: Observable vs. reconstructible vs. detectable
• $f(x, y) = x^2 + 3y$
  Terms: partial derivative; gradient; Jacobian; directional derivative; differential; Lie derivative
  Distinctions: Types of multivariable derivatives
• $x \mapsto \sin(1/x)$ on $(0, 1]$
  Terms: continuous; continuously differentiable; locally Lipschitz continuous; uniformly continuous; Lipschitz continuous
  Distinctions: Types of continuity
• $x \mapsto \sqrt{x}$ on $[0, 1]$
  Terms: uniformly continuous; absolutely continuous; continuous; locally Lipschitz continuous; Lipschitz continuous; continuously differentiable
  Distinctions: Types of continuity
• $x \mapsto \abs{x_1} + \abs{x_2}$ on $\reals^2$
  Terms: norm; inner product
  Distinctions: Norm vs. metric vs. inner product
• $x \mapsto x^3$, from $\reals$ to $\reals$
  Terms: continuous; open map
• $(x, y) \mapsto 2 x_1 y_1 + x_2 y_2$ on $\reals^2$
  Terms: bilinear form; inner product; dot product
  Distinctions: Dot product vs. inner product vs. bilinear form
• $(x, y) \mapsto x_1 y_1 - x_2 y_2$ on $\reals^2$
  Terms: bilinear form; inner product
  Distinctions: Dot product vs. inner product vs. bilinear form
• $(x, y) \mapsto x_1 y_1 + x_2 y_2$ on $\reals^2$
  Terms: bilinear form; inner product; dot product
  Distinctions: Dot product vs. inner product vs. bilinear form; Norm vs. metric vs. inner product
• $\dot{x} = -x$, $y = 0$
  Terms: detectable; observable; reconstructible
  Distinctions: Observable vs. reconstructible vs. detectable
• $\dot{x} = x$, $y = x$
  Terms: observable; reconstructible; detectable
  Distinctions: Observable vs. reconstructible vs. detectable
• A function with $f^{\prime}(x) = 0$ for every $x$ in an interval is constant on the interval, as a consequence of the mean value theorem

Graduate

• $[0, 1]$, for $\dot{x} = u$ with $\abs{u} \leq 1$
• A “fat” Cantor set
  Terms: nowhere dense; measure zero
  Distinctions: Types of set sparsity
• A point under rotation of a circle by the angle $2\pi\sqrt{2}$
  Terms: periodic
• An unscented Kalman filter
  Terms: estimator; filter (estimation); observer; predictor
  Distinctions: Types of estimators
• The Borel sets of $\reals$

  The smallest σ-algebra on $\reals$ that contains every open interval.

  Terms: σ-algebra; topology
  Distinctions: Topology vs. σ-algebra
• Continuous functions on $[0, 1]$ with the max norm
  Terms: metric space; Banach space
• The first-order sentence $\forall x \, (x^2 \geq 0) \land \neg \exists x \, (x^2 < 0)$, whose propositional form is $p \land \neg q$
  Terms: sentence; propositional formula
  Distinctions: Sentence vs. propositional formula
• The subsets of $\reals$ that are finite or have a finite complement
  Terms: σ-algebra; topology
  Distinctions: Topology vs. σ-algebra
• Minimizing $\int_0^\infty \bigl(x(t)^2 + u(t)^2\bigr) \, \mathrm{d}t$ for the system $\dot{x} = x + u$, with $x(0) = 1$
  Terms: calculus of variations; optimal control
  Distinctions: Calculus of variations vs. optimal control
• $(0, 1)$ and $(1, 2)$
• A polar decomposition $A = Q S$, with $Q$ orthogonal and $S$ symmetric positive semidefinite
  Terms: QR decomposition
  Distinctions: Types of matrix decompositions
• Polynomials on $[0, 1]$ with the max norm
  Terms: metric space; Banach space
• $r \mapsto \arctan r$ on $[0, \infty)$
  Terms: class-$\mathcal{K}$; positive definite function; class-$\mathcal{K}_\infty$
• The class of all sets
  Terms: collection; set
  Distinctions: Types of collections
• The closed unit ball of $\ell^2$
• The line with two origins
  Terms: T₁